zbMATH — the first resource for mathematics

The Gauss section of a Riemannian immersion. (English) Zbl 0607.53036
Given a Riemannian immersion \(\phi\) : \(M\to N\), let \(m=\dim M\) and let \(G_ mN\) be the Grassmann bundle of m planes tangent to N. Then \(\phi\) induces the Gauss section \(\gamma_{\phi}: M\to G_ mN\) by the formula \(\gamma_{\phi}(x)=d\phi (T_ xM)\). Note that \(\gamma_{\phi}\) may be viewed as a section of the bundle \(\phi^{-1}(G_ mN).\)
The author studies \(\gamma_{\phi}\) in the context of sections of a submersion \(\pi\) : \(P\to M\) of Riemannian manifolds. The vertical energy of a map \(\sigma\) : \(M\to P\) is obtained from the vertical component of \(d\sigma\), and the interesting variations are those tangent to the level manifolds of \(\pi\). There is a vertical tension which vanishes if and only if \(\sigma\) is stationary under such variations (i.e. vertically harmonic in the author’s terminology). The equations of Gauß and Codazzi for the immersion give rise to conditions for vertical conformality and vertical harmonicity of the Gauss section, which are also related to a Weitzenböck formula. Various geometric consequences are drawn, generalizing known results for N of constant curvature.
Reviewer: B.L.Reinhart

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E20 Harmonic maps, etc.
Full Text: DOI